Indeed it is a speckle pattern, and despite LEDs not being coherent sources, the spacial coherence can still be enough to observe such effects -- thanks to their intensity and small size, they make good point sources.
I've observed this with my strings of, uh, whatever they were, 7x7mm (4-pin THT) red superbright LEDs. Presumably the speckle is characteristic of the component lens, the air between (mostly irrelevant at indoor conditions), and your eyes. So it varies as you move around, and seems to track your eyes say as you rotate your head.
You can see similar patterns in the bokeh when defocused, or viewing through a magnifier or such. (I feel a loupe/microscope shows these fairly often, as the depth-of-field is quite poor, and we often work with strongly specular materials e.g. "bright" solder joints and flux, glossy plastic parts, solder mask, etc.).
Also contributes (or contributed by) normal eye wear/aging; I've noticed a little bit of, astigmatism I suppose, in the last I don't know five years or so; bright sources (particularly at night, both because of contrast and wide iris) having more of a halo, plus some streaky shapes which can be shadowed in certain regions (e.g. hold my finger near my eye, partially shadowing the light source from one side or another). Well, I was told a long time ago that I probably wouldn't need glasses until I was 40, and there's a few years left before that, so that seems on track.
Tim